Similar to Theorem
4.4, we write
$$ f(y)=\sum^{\infty}_{j=-\infty}f(y) \chi_{E_{j}}(y)=\sum^{\infty }_{j=-\infty}f_{j}(y). $$
Hence, we have
$$\begin{aligned} \bigl\Vert [b,R_{L}]f\bigr\Vert _{\dot{K}_{q}^{\alpha,p}(\mathbb{R}^{n})}^{p} \lesssim& \sum^{\infty}_{k=-\infty}2^{k\alpha p} \Biggl(\sum^{k-2}_{j=-\infty}\bigl\Vert \chi_{k}[b,R_{L}]f_{j}\bigr\Vert _{L^{q}(\mathbb {R}^{n})} \Biggr)^{p} \\ &{}+ \sum^{\infty}_{k=-\infty}2^{k\alpha p} \Biggl(\sum^{k+1}_{j=k-1}\bigl\Vert \chi _{k}[b,R_{L}]f_{j}\bigr\Vert _{L^{q}(\mathbb{R}^{n})} \Biggr)^{p} \\ &{}+ \sum^{\infty}_{k=-\infty}2^{k\alpha p} \Biggl(\sum^{\infty }_{j=k+2}\bigl\Vert \chi_{k}[b,R_{L}]f_{j}\bigr\Vert _{L^{q}(\mathbb{R}^{n})} \Biggr)^{p} \\ =& \widetilde{W}_{1}+\widetilde{W}_{2}+ \widetilde{W}_{3}. \end{aligned}$$
For
\(\widetilde{W}_{2}\), by Lemma
4.3, we have
$$\widetilde{W}_{2}\lesssim\sum^{\infty}_{k=-\infty}2^{k\alpha p} \Biggl(\sum^{k+1}_{j=k-1}\|f_{j} \|_{L^{q}(\mathbb{R}^{n})} \Biggr)^{p}\lesssim \|f\|_{K_{q}^{\alpha,p}(\mathbb{R}^{n})}^{p}. $$
Now we estimate
\(\widetilde{W}_{1}\). For
\(x\in E_{k}\),
\(y\in E_{j}\) with
\(j\leq k-2\),
\(|x-y|\sim2^{k}\). By Lemma
2.16, we can get
$$\begin{aligned}& \bigl\Vert \chi_{k}[b,R_{L}]f_{j}\bigr\Vert _{L^{q}(\mathbb{R}^{n})} \\& \quad = \biggl( \int_{E_{k}}\biggl\vert \int_{E_{j}} \bigl(b(x)-b(y) \bigr)K_{R_{L}}(x,y)f(y)\,dy \biggr\vert ^{q}\,dx \biggr)^{\frac{1}{q}} \\& \quad \lesssim \biggl( \int_{E_{k}}\biggl\vert \int_{E_{j}}\frac{| (b(x)-b(y) )f(y)|}{(1+|x-y|/\rho(x))^{N}}\frac{1}{|x-y|^{n-1}} \int_{B(y,|x-y|)}\frac{V(z)}{|z-y|^{n-1}}\,dz\,dy\biggr\vert ^{q}\,dx \biggr)^{\frac {1}{q}} \\& \qquad {}+ \biggl( \int_{E_{k}}\biggl\vert \int_{E_{j}}\frac{| (b(x)-b(y) )f(y)|}{(1+|x-y|/\rho(x))^{N}}\frac{1}{|x-y|^{n}} \,dy\biggr\vert ^{q}\,dx \biggr)^{\frac{1}{q}} \\& \quad = \widetilde{W}_{1,1}+\widetilde{W}_{1,2}. \end{aligned}$$
For
\(\widetilde{W}_{1,1}\), we can see that
\(b(x)-b(y)=(b(x)-b_{B_{j}})-(b(y)-b_{B_{j}})\). Hence, by Lemma
2.17, we have
$$\begin{aligned} \widetilde{W}_{1,1} \lesssim& \frac{1}{(1+2^{k}/\rho(0))^{N/l_{0}+1}} \\ &{}\times\frac {1}{|2^{k}|^{n-1}} \biggl( \int_{E_{k}}\biggl\vert \int_{E_{j}}\bigl\vert \bigl(b(x)-b(y) \bigr)f(y)\bigr\vert \int_{B(y,|x-y|)}\frac{V(z)}{|z-y|^{n-1}}\,dz\,dy\biggr\vert ^{q}\,dx \biggr)^{\frac {1}{q}} \\ \lesssim& \frac{1}{(1+2^{k}/\rho(0))^{N/l_{0}+1}}\frac{1}{|2^{k}|^{n-1}} \biggl( \int _{E_{k}}\bigl\vert \bigl(b(x)-b_{B_{j}} \bigr) \bigr\vert ^{q}\,dx \biggr)^{\frac{1}{q}} \\ &{}\times \int_{E_{j}}\bigl\vert f(y)\bigr\vert \int_{B(0,2^{k+1})}\frac{V(z)}{|z-y|^{n-1}}\,dz \,dy \\ &{}+ \frac{1}{(1+2^{k}/\rho(0))^{N/l_{0}+1}}\frac{|E_{k}|^{\frac {1}{q}}}{|2^{k}|^{n-1}} \int_{E_{j}}\bigl\vert \bigl(b(y)-b_{B_{j}} \bigr)f(y) \bigr\vert \\ &{}\times \int_{B(0,2^{k+1})}\frac{V(z)}{|z-y|^{n-1}}\,dz \,dy \\ =& \widetilde{W}_{1,1}^{(1)}+\widetilde{W}_{1,1}^{(2)}. \end{aligned}$$
Using Hölder’s inequality and the boundedness of the fractional integral
\(\mathcal{I}_{1}\) with
\(\frac{1}{\tilde{q}}=\frac {1}{s}-\frac {1}{n}\), we obtain
$$\begin{aligned}& \int_{E_{j}}\bigl\vert \bigl(b(y)-b_{B_{j}} \bigr)f(y) \bigr\vert \int _{B(0,2^{k+1})}\frac {V(z)}{|z-y|^{n-1}}\,dz\,dy \\& \quad \lesssim\frac{\|b\|_{\sigma}|E_{j}|^{1-\frac{1}{q}-\frac {1}{\tilde {q}}}}{(1+2^{j}/\rho(0))^{-(l_{0}+1)\sigma}}\|f_{j}\|_{L^{q}(\mathbb{R}^{n})} \bigl\Vert \mathcal{I}_{1}(V\chi_{B(0,2^{k+1})})\bigr\Vert _{L^{\tilde{q}}(\mathbb {R}^{n})} \\& \quad \lesssim\frac{\|b\|_{\sigma}|E_{j}|^{1-\frac{1}{q}-\frac {1}{\tilde {q}}}}{(1+2^{j}/\rho(0))^{-(l_{0}+1)\sigma}}\|f_{j}\|_{L^{q}(\mathbb{R}^{n})} \|V \chi_{B(0,2^{k+1})}\|_{L^{s}(\mathbb{R}^{n})} \\& \quad \lesssim \frac{\|b\|_{\sigma}\|f_{j}\|_{L^{q}(\mathbb{R}^{n})}}{(1+2^{k}/\rho (0))^{-(l_{0}+1)\sigma-N_{0}}} |E_{j}|^{1-\frac{1}{q}-\frac{1}{\tilde{q}}}|E_{k}|^{\frac {1}{s}-\frac{2}{n}}, \end{aligned}$$
where in the last inequality we have used (
4.2) above. We get
$$ \widetilde{W}_{1,1}^{(2)} \lesssim \frac{\|b\|_{\sigma}\|f_{j}\| _{L^{q}(\mathbb{R}^{n})}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)-(l_{0}+1)\sigma -N_{0}}} \frac{|E_{j}|^{1-\frac{1}{q}-\frac{1}{\tilde {q}}}}{|E_{k}|^{1-\frac{1}{q}-\frac{1}{\tilde{q}}}}. $$
Similar to
\(\widetilde{W}_{1,1}^{(2)}\), we have
$$ \int_{E_{j}}\bigl\vert f(y)\bigr\vert \int_{B(0,2^{k+1})}\frac{V(z)}{|z-y|^{n-1}}\,dz\,dy \lesssim \|f_{j}\|_{L^{q}(\mathbb{R}^{n})}\bigl(1+2^{k}/\rho(0) \bigr)^{N_{0}} |E_{j}|^{1-\frac{1}{q}-\frac{1}{\tilde{q}}}|E_{k}|^{\frac {1}{s}-\frac{2}{n}}. $$
By Lemma
2.10, we get
$$ \widetilde{W}_{1,1}^{(1)} \lesssim \frac{(k-j)\|b\|_{\sigma}\| f_{j}\| _{L^{q}(\mathbb{R}^{n})}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)-(l_{0}+1)\sigma -N_{0}}} \frac{|E_{j}|^{1-\frac{1}{q}-\frac{1}{\tilde {q}}}}{|E_{k}|^{1-\frac{1}{q}-\frac{1}{\tilde{q}}}}. $$
Since
N is large enough, when
\(j\leq k-2\), we have
$$ \widetilde{W}_{1,1}\lesssim (k-j)2^{(j-k)n(1-\frac{1}{q}-\frac{1}{\tilde{q}})}\|b\|_{\sigma} \| f_{j}\|_{L^{q}(\mathbb{R}^{n})}. $$
Next we estimate
\(\widetilde{W}_{12}\). We can get
$$\begin{aligned} \widetilde{W}_{1,2} \lesssim& \frac{1}{(1+2^{k}/\rho(0))^{N/(l_{0}+1)}}\frac{1}{|2^{k}|^{n}} \biggl( \int _{E_{k}}\bigl\vert \bigl(b(x)-b_{B_{j}} \bigr) \bigr\vert ^{q}\,dx \biggr)^{\frac{1}{q}} \int_{E_{j}}\bigl\vert f(y)\bigr\vert \,dy \\ &{}+ \frac{1}{(1+2^{k}/\rho(0))^{N/(l_{0}+1)}}\frac{|E_{k}|^{\frac {1}{q}}}{|2^{k}|^{n}} \int_{E_{j}}\bigl\vert \bigl(b(y)-b_{B_{j}} \bigr)f(y) \bigr\vert \,dy \\ =: &\widetilde{W}_{1,2}^{(1)}+\widetilde{W}_{1,2}^{(2)}. \end{aligned}$$
Using Lemma
2.10 and the Hölder inequality, we get the following estimates:
$$\widetilde{W}_{1,2}^{(1)}\lesssim\frac{(k-j)\|b\|_{\sigma}\|f_{j}\| _{L^{q}(\mathbb{R}^{n})}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)-(l_{0}+1)\sigma}} \frac{|E_{j}|^{1-\frac{1}{q}}}{|E_{k}|^{1-\frac{1}{q}}} $$
and
$$\widetilde{W}_{1,2}^{(2)}\lesssim \frac{\|b\|_{\sigma}\|f_{j}\|_{L^{q}(\mathbb{R}^{n})}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)-(l_{0}+1)\sigma}} \frac{|E_{j}|^{1-\frac{1}{q}}}{|E_{k}|^{1-\frac{1}{q}}}. $$
Take
N large enough. Because
\(j\leq k-2\), we can see that
$$\widetilde{W}_{1, 2}\lesssim(k-j)2^{(j-k)n(1-\frac{1}{q}-\frac {1}{\tilde{q}})}\|b\|_{\sigma} \|f_{j}\|_{L^{q}(\mathbb{R}^{n})}. $$
Hence,
$$\bigl\Vert \chi_{k}[b,R_{L}]f_{j}\bigr\Vert _{L^{q}(\mathbb{R}^{n})}\lesssim (k-j)2^{(j-k)n(1-\frac{1}{q}-\frac{1}{\tilde{q}})}\|b\|_{\sigma}\| f_{j}\|_{L^{q}(\mathbb{R}^{n})}. $$
Since
\(-\frac{n}{q}<\alpha<n(1-\frac{1}{q}-\frac{1}{\tilde{q}})\), similar to Theorem
3.10, we use Hölder’s inequality to obtain
$$\begin{aligned} \widetilde{W}_{1} \lesssim& \sum^{\infty}_{k=-\infty}2^{k\alpha p} \Biggl(\sum^{k-2}_{j=-\infty}(k-j)2^{(j-k)n(1-\frac{1}{q}-\frac{1}{\tilde {q}})} \|b\|_{\sigma}\|f_{j}\|_{L^{q}(\mathbb{R}^{n})} \Biggr)^{p} \\ \lesssim& \|b\|_{\sigma}^{p}\|f\|^{p}_{\dot{K}_{q}^{\alpha, p}}. \end{aligned}$$
For
\(\widetilde{W}_{3}\), if
\(x\in E_{k}\) and
\(y\in E_{j}\) with
\(j\geq k+2\), then
\(|x-y|\sim2^{j}\). We have
$$\begin{aligned}& \bigl\Vert \chi_{k}[b,R_{L}]f_{j}\bigr\Vert _{L^{q}(\mathbb{R}^{n})} \\& \quad \lesssim \biggl( \int_{E_{k}}\biggl\vert \int_{E_{j}}\frac{| (b(x)-b(y) )f(y)|}{(1+|x-y|/\rho(x))^{N}}\frac{1}{|x-y|^{n-1}} \int_{B(y,|x-y|)}\frac{V(z)}{|z-y|^{n-1}}\,dz\,dy\biggr\vert ^{q}\,dx \biggr)^{\frac {1}{q}} \\& \qquad {}+ \biggl( \int_{E_{k}}\biggl\vert \int_{E_{j}}\frac{| (b(x)-b(y) )f(y)|}{(1+|x-y|/\rho(x))^{N}}\frac{1}{|x-y|^{n}} \,dy\biggr\vert ^{q}\,dx \biggr)^{\frac{1}{q}} \\& \quad =: \widetilde{W}_{3,1}+\widetilde{W}_{3,2}. \end{aligned}$$
Using the decomposition
\(b(x)-b(y)=(b(x)-b_{B_{k}})-(b(y)-b_{B_{k}})\) again, we get
$$\begin{aligned}& \int_{E_{j}}\bigl\vert \bigl(b(y)-b_{B_{k}} \bigr)f(y) \bigr\vert \int_{B(y,|x-y|)}\frac {V(z)}{|z-y|^{n-1}}\,dz \\& \quad \lesssim \int_{E_{j}}\bigl\vert \bigl(b(y)-b_{B_{k}} \bigr)f(y) \bigr\vert \int _{B(0,2^{j+1})}\frac{V(z)}{|z-y|^{n-1}}\,dz\,dy \\& \quad \lesssim \frac{(k-j)\|b\|_{\sigma}\|f_{j}\|_{L^{q}(\mathbb {R}^{n})}}{(1+2^{j}/\rho(0))^{-(l_{0}+1)\sigma-N_{0}}} |E_{j}|^{1-\frac{1}{q}-\frac{1}{\tilde{q}}}|E_{j}|^{\frac {1}{s}-\frac{2}{n}}. \end{aligned}$$
On the other hand,
$$ \int_{E_{j}}\bigl\vert f(y)\bigr\vert \int_{B(0,2^{j+1})}\frac{V(z)}{|z-y|^{n-1}}\,dz\,dy \lesssim \|f_{j}\|_{L^{q}(\mathbb{R}^{n})}\bigl(1+2^{j}/\rho(0) \bigr)^{N_{0}} |E_{j}|^{1-\frac{1}{q}-\frac{1}{\tilde{q}}}|E_{j}|^{\frac {1}{s}-\frac{2}{n}}. $$
Therefore
$$ \widetilde{W}_{3,1}\lesssim (j-k)2^{(k-j)n(\frac{1}{q})}\|b\|_{\sigma} \|f_{j}\|_{L^{q}(\mathbb{R}^{n})}. $$
Similarly, we can get
$$ \widetilde{W}_{3,2}\lesssim (j-k)2^{(k-j)n(\frac{1}{q})}\|b\|_{\sigma} \|f_{j}\|_{L^{q}(\mathbb{R}^{n})}. $$
Because
\(-\frac{n}{q}<\alpha<n(1-\frac{1}{q}-\frac{1}{\tilde {q}})\), by the estimates for
\(\widetilde{W}_{31}\) and
\(\widetilde{W}_{33}\), we can use Hölder’s inequality and (
4.1) to get
$$ \widetilde{W}_{3} \lesssim \sum^{\infty}_{k=-\infty}2^{k\alpha p} \Biggl(\sum^{k-2}_{j=-\infty}(j-k)2^{(k-j)n(\frac{1}{q})} \|b\|_{\sigma}\| f_{j}\|_{L^{q}(\mathbb{R}^{n})} \Biggr)^{p} \lesssim\|b\|_{\sigma}^{p}\|f\|^{p}_{\dot{K}_{q}^{\alpha,p}}. $$
This completes the proof of Theorem
4.5. □